Plane Partitions Fermion-Boson Duality

Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality

Thiago Araujo,1, ∗ Domenico Orlando,1, 2, † and Susanne Reffert1, ‡ 1Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, ch-3012, Bern, Switzerland 2INFN, sezione di Torino and Arnold-Regge Center, via Pietro Giuria 1, 10125 Torino, Italy In this work we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal melting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions, and writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, based upon the latter result we conjecture that the growth operators for the quantum Hamiltonian can be represented in terms of the affine Yangian Y[glb(1)].

I. INTRODUCTION addressed in [9]. Plane partitions can be written in terms of interlaced XXZ diagrams or as lattice fermions, see Random partitions appear in many contexts in mathe- figure (2). The empty partition is a configuration of the matics and physics. This omnipresence is partly explained hexagonal lattice where all positions below the diagonals by the fact that they are part of the core of number theory, are occupied. Additionally, based on the one- and two- the queen of mathematics [1]. It is nonetheless remarkable dimensional cases, a conjecture for the mass gap has been that they appear in a variety of distinct problems – such made, and further numerical analysis has been performed. as the Seiberg–Witten theory, integrable systems, black We present a number of equivalent formulations of the holes, string theory and others [2–6]. 3D quantum crystal melting problem, such as in terms of In two dimensions, random partitions can be successfully particle-hole hopping, a fermionic description on a Kagome realized in terms of fermionic operators living on a chain lattice, and a tensor product representation. We are even- where particles and holes are labeled by half-integers along tually lead to a fermion-boson duality, and to a conjecture the real line [3, 7–9]. In this formalism, the empty partition for an underlying Yangian symmetry. We expect this is equivalent to a configuration where all negative holes (or structure to be instrumental in uncovering a possible hid- better yet, holes on negative positions) are occupied and den integrable structure, suggested by the results in lower all positive holes are vacant. The creation and annihilation dimensions. The plan of this note is as follows. We start with a short review in SectionII, where we see how plane partitions are equivalently written in terms of dimers in a hexagonal lattice.

FIG. 1: Fermionic ﬁelds and Young diagram relation. arXiv:2005.09103v2 [hep-th] 27 Jan 2021

of squares are defined as holes and particles hopping as in figure (1). FIG. 2: 3D Young diagram and its fermions lattice The description above has been used in [9, 10] to study the two-dimensional version of quantum crystal melting. Quite remarkably, with a Jordan-Wigner transformation, As we said before, we want to find an alternative particle- the crystal melting Hamiltonian is shown to be equivalent hole dynamics which is equivalent to the plane partition to the XXZ spin chains with kink boundary conditions, and growth. More explicitly, we search for a statistical sys- it automatically implies the integrability of this problem. tem with states labeled by plane partitions. That is the problem we address in SectionIII, where we discuss a In this work, we focus on the three-dimensional version construction in terms of particle hopping in a hexagonal of the quantum crystal melting, which can be rephrased in lattice. This formalism is considerably simplified if we terms of random plane partitions, and it has been partly use a dual description which turns out to be the Kagome lattice. We write the new Hamiltonian, and we provide additional details of it in SectionIV. ∗Electronic address: [email protected] Having discussed characterizations of the crystal melting †Electronic address: [email protected] as occupation problems, we try to understand other fun- ‡Electronic address: sreff[email protected] damental aspects of the system, in particular, its Hilbert 2 space. In SectionV, we write the plane partition states that there is only a finite number of boxes that can be in terms of interlaced integer partitions, and we use this consistently removed, or available places where we can add fact to show that the growth operators have nontrivial a box. coproducts. In addition, we define a novel fermion-boson correspondence for plane partitions states in SectionVI. Finally, based on the action of the growth operators on the bosonized states, we conjecture that this Hamiltonian can represented in terms of the affine Yangian of glb(1). We give some concluding remarks in SectionVII. In AppendixA, we give present some necessary details on the 2D fermion- boson duality, in AppendixB we argue the equivalence of the Hamiltonians in the plane-partition language and in the Kagome picture. In AppendixC, we detail basis transformations for various states and in AppendixD, we give amplitudes for transitions between various states. FIG. 4: 1 Box configuration

II. PLANE PARTITIONS & HEXAGONAL We want to study the crystal melting Hamiltonian de- DIMERS fined in [9] X We start by describing the well known correspondence H = − J (| ih | + | ih |) between the crystal melting problem and dimers in a hexag- X √ 1 (2a) + V q| ih | + √ | ih | , onal lattice, see [9, 11–13] and references therein. The q empty partition (vacuum) is equivalent to an infinity per- fect matching with different staggered boundary conditions that we conveniently write as in the regions I, II and III, see figure (3), defined as √ 1 H = − J (Γ + Π) + V q O (Π, Γ) + √ O (Π, Γ) . ϑ ∈ I = (−π/6, π/2), + q − ϑ ∈ II = (π/2, 7π/6), (1) (2b) ϑ ∈ III = (−5π/6, −π/6). Using | i† = h | and | i† = h |, one can easily see In the vacuum configuration, the origin is defined as that the Hamiltonian is Hermitian. The two terms multi- the only plaquette where the three phases, or staggered plied by J are kinetic and describe the two flips previously directions, meet. We say that a plaquette with the three mentioned. More specifically, the operator Γ (Π) adds phases is a flippable plaquette. (removes) boxes in all possible ways. Moreover, the potential terms O+(Π, Γ) and O−(Π, Γ) count the number of positive and negative flippable plaquettes, respectively. In other words, they determine the number of places where we can add a box, and the number of boxes that can be removed respecting the plane partition rules. This system has a special point for J = V , the so-called Rokhsar-Kivelson point [14, 15], where it is natural to understand the model as the stochastic quantization of the classical counting of plane partitions [16]. In the continuum limit this corresponds to a quantum critical point [17]. At this point we expect the system to have special symmetry FIG. 3: Empty partition properties and we will focus on this case. Some progress in the understanding of this system has been achieved in [9], where it was shown that the ground state of the system is Adding a box to the vacuum corresponds to the positive given by a sum over all plane partitions, where a partition flip ( ) 7→ ( ), as seen in figure (4), and as a matter of of the integer k is weighted by qk/2. If follows that the fact, there is only one flippable plaquette in the vacuum. norm of the ground state is given by Additionally, simple inspection of figure (4) reveals that there are now three plaquettes where we can perform a Y 1 Z ≡ hground|groundi = , (3) positive flip, and just one plaquette where the negative flip (1 − qn)n ( ) 7→ ( ) can be performed. n≥1 It is easy to see that this logic remains the same for general configurations, that is, positive and negative flips which we recognize as the famous MacMahon function: the correspond to adding and removing boxes in a given plane generating function for the number pl(k) of plane partitions partition state. Furthermore, for any lattice configuration, ∞ X there is only a finite number of flippable plaquettes, and Z = pl(k)qk . (4) from the plane partition perspective it obviously means k=0 3

In the following sections, we want to represent the Hamil- Furthermore, we set the origin at (j, k) = ~0, and we can 2 tonian (2b) in terms of a statistical mechanics systems of identify a generic point ~x = j~p1 + k~p2,(j, k) ∈ Z as in particles and holes, and we start investigating its integra- figure (6). bility properties. More specifically, we define a new form of fermion-boson correspondence and we find connections between this problem and the affine Yangians of glb(1).

III. PARTICLE-HOLE BOUND STATES

In this section we describe the dynamics of the crystal melting problem in terms of particle hopping in two Bravais FIG. 6: Generic points in the lattice. lattices (2b). In the next section we build a Hamiltonian for this problem using the dual description of this particle-hole Points in the Hole sublattice H can be identified by linear system. combinations of (~p1, ~p2, ~p3) plus the additional offsets First we need to remember that the hexagons above do not define a Bravais lattice, but we can write it as ~ a 1 ~ 1 δ± = √ , ±1 , δ = a −√ , 0 . (9) a superposition of two triangular Bravais lattices, say H 2 3 3 whose points are labeled by ◦, and P, labeled by •. We call them Hole and Particle sublattices, respectively. Putting all these facts together, we can try to build a The position vectors in the P sublattice are model where the flips in the hexagonal dimers correspond to simultaneous hoppings in the two sublattices. a √ a √ ~p1 = ( 3, 1), ~p2 = ( 3, −1), (5) The creation of one box is described as a counterclock- 2 2 wise rotation as in figure (7), and it is equivalent to three where a is the lattice spacing, that can, without loss of simultaneous hoppings, a 3-hopping, along the positively generality, be set to a = 1, see details in [18]. In figure (5) oriented (the red arrow) hole-particle bound states, we have also given the linear combination ~p3 = ~p1 + ~p2. ∗ ~ Finally, the dimer between two lattice points is represented | i =ψ (~p1 + δ)ψ(~p1) (10a) by the positively oriented bound states ◦ → •. ∗ ~ ∗ ~ ψ (~p2 + δ)ψ(~0)ψ (~p1 + ~p2 + δ)ψ(~p2)|∅i.

FIG. 7: Counterclockwise 3-hopping FIG. 5: Sublattices H (◦) and P (•) This point of view describes the box creation in terms of The fermionic operators ψ(~x) and ψ∗(~x) in an arbitrary next-neighbor interactions, but it may be computationally lattice point ~x satisfy the anticommutation relations useful to describe the box creation in terms of the horizontal 3-hopping that we define as in figure (8). {ψ(~x), ψ(~x0)} = 0, {ψ∗(~x), ψ∗(~x0)} = 0, (6) ∗ 0 {ψ (~x), ψ(~x )} = δ~x,~x0 . The bottom of the Dirac sea is defined by the condition

ψ(~x)|0i = 0 , ∀ ~x ∈ H, P , (7) FIG. 8: Horizontal 3-hopping and the empty partition (vacuum) by

Y Y ∗ In other words, we reshuffle the expression (10a) |∅i = ψ (j~p1 + k~p2)|0i . (8) j∈Z k∈Z ∗ ~ ∗ ~ | i =ψ (~p2 + δ)ψ(~p2)ψ (~p1 + ~p2 + δ)ψ(~0) (10b) ∗ ~ With these definitions, it is straightforward to see that all ψ (~p1 + δ)ψ(~p1)|∅i . points in the sublattice P are occupied. It should be implicit (since it is a visual aid) that the holes and particles are The 3-hopping above describes the creation and annihi- bonded positively, ◦ → •. A flip in the hexagon (creation lation of a state corresponding to a partition with one box. or annihilation of a box) is simply a hole-particle hopping For configurations with more boxes, some of the relevant which preserves the positive orientation of the bound state. bound states are edges of two hexagons. In this case, we 4 would need to consider a horizontal 2-hopping where there are two free bound states, and the usual hopping when there is just one free bound state. In other words, we write the 3 configurations with 2 boxes as

E ∗ ~ =ψ (2~p1 + δ)ψ(~p1 − ~p2) (11a) ∗ ~ FIG. 10: Empty conﬁguration in the Kagome lattice ψ (2~p1 − ~p2 + δ)ψ(2~p1 − ~p2)| i ,

∗ ~ | i =ψ (2~p2 + δ)ψ(~p1 + ~p2) where |0˜i is the Dirac sea in the Kagome lattice, which is (11b) deﬁned by putting a dual hole at the midpoint of each hon- ψ∗(~p + 2~p + ~δ)ψ(2~p )| i, 1 2 2 eycomb edge, or equivalently, in the {3, 6, 3, 6} tessellation, and see ﬁgure (11). ∗ ~ | 2 i =ψ (−~p1 + ~p2 + δ)ψ(−~p1) (11c) ∗ ~ ψ (δ)ψ(−~p1 + ~p2)| i . This description is rather involved, and somewhat con- trived, for states with more than one box. Fortunately, in the next section we show that there exists a more natural description when we assume that for each dimer, there is a dual description in which this extended object is pointlike. FIG. 11: Dual Dirac sea.

Moreover, we have defined IV. DUAL KAGOME LATTICE ∆ = ξ~ δ [arg(j ~m + k ~m )]+ In order to define this dual description, let us start with j,k + I 1 2 ~ ~ the basis ξ−δII[arg(j ~m1 + k ~m2)] + ξδIII[arg(j ~m1 + k ~m2)] , a √ (14b) ~m = a(0, 1) , ~m = ( 3, −1) , (12) 1 2 2 where connecting the barycenters of hexagons, as in figure (9). ( Furthermore, the offset vectors from the center of a hexagon 0 if arg(~v) ∈/ A δ [arg(~v)] = , A = I, II, III , (14c) A 1 if arg(~v) ∈ A

and the regions I, II and III have been defined in Eq. (1), see also figure (3). Moreover, it is easy to write an explicit expression for δA[arg(~v)] as a difference of Heaviside functions. Using the dual lattice, the box creation and annihilation are given by the simultaneous action of three pairs of dual hole-particle pairs ψ˜∗ψ˜ at neighboring vertices. The vectors from the origin to the vertices are given by FIG. 9: Dual basis ~ (s) to its edges are Pj,k = j ~m1 + k ~m2 a a √ a (15) ξ~ = (± 3, 1) , ξ~ = (0, −1) . (13) + (cos((2s + 1)π/6), sin((2s + 1)π/6)), ± 4 2 2 s = 0, 1, 2, 3, 4, 5 , Given a hexagon in the lattice, let us associate a dual ˜∗ fermion (created by ψ ) to the midpoint of each edge where we have introduced an index s as in figure (12). containing a dimer and a dual hole (where the fermion is annihilated by ψ˜) to the midpoint of each empty edge. Connecting these midpoints generates another lattice made of hexagons and triangles, which is known as Archimedean tessellation {3, 6, 3, 6} [19], or Kagome lattice. The empty configuration corresponds to figure (10). In terms of the dual fermions, the empty configuration is given by Y Y |∅i = ψ˜∗(j ~m + k ~m + ∆ )|0˜i , (14a) 1 2 j,k FIG. 12: Vector position in the dual lattice j∈Z k∈Z 5

Putting all these facts together, the 1-box conﬁguration Observe that the action of the operator Γj,k on |∅i for any is given by position j 6= 0 and k 6= 0 naturally vanishes because there is a mismatch of fermions and holes. | i =ψ˜∗(P~ (1))ψ˜(P~ (0))ψ˜∗(P~ (3))ψ˜(P~ (2)) 0,0 0,0 0,0 0,0 (16a) ˜∗ ~ (5) ˜ ~ (4) As the notation above suggests, we represent the opera- ψ (P0,0 )ψ(P0,0 )|∅i tors Γ and Π in (2b) as with inverse

|∅i =ψ˜∗(P~ (0))ψ˜(P~ (1))ψ˜∗(P~ (2))ψ˜(P~ (3)) 0,0 0,0 0,0 0,0 X X ˜(1)∗ ˜(0) ˜(3)∗ ˜(2) ˜(5)∗ ˜(4) (16b) Γ = Γj,k = ψ ψ ψ ψ ψ ψ (18a) ˜∗ ~ (4) ˜ ~ (5) j,k j,k j,k j,k j,k j,k ψ (P0,0 )ψ(P0,0 )| i . j,k j,k∈Z X X ˜(0)∗ ˜(1) ˜(2)∗ ˜(3) ˜(4)∗ ˜(5) ˜(s) ˜ ~ (s) Π = Πj,k = ψj,k ψj,k ψj,k ψj,k ψj,k ψj,k . (18b) Making the notation lighter with ψj,k ≡ ψ(Pj,k ), we have j,k j,k∈Z X ˜(1)∗ ˜(0) ˜(3)∗ ˜(2) ˜(5)∗ ˜(4) | i = ψj,k ψj,k ψj,k ψj,k ψj,k ψj,k |∅i (17a) j,k∈Z One may also calculate the commutators among the com- X ≡ Γj,k|∅i ponents Γij and Πij of the growth operators. It is easy j,k∈Z to see that the only nontrivial commutator is [Γij, Πkl]. X (0)∗ (1) (2)∗ (3) (4)∗ (5) Although it does not give any new information, it is useful |∅i = ψ˜ ψ˜ ψ˜ ψ˜ ψ˜ ψ˜ | i (17b) j,k j,k j,k j,k j,k j,k to have its explicit expression since it is important in our j,k∈Z calculations. It is straightforward (but laborious) to check X ≡ Πj,k| i . the following relation: j,k∈Z

(54¯ 32¯ 1)¯ (123¯ 45)¯ (0)¯ (54¯ 32)¯ (23¯ 45)¯ (0) (01)¯ (54¯ 3)¯ (345)¯ (10)¯ [Γij, Πkl] = δikδjl ψij ψkl − ψkl ψij ψkl ψij + ψkl ψij ψkl ψij (19) (01¯ 2)¯ (54)¯ (45)¯ (210)¯ (01¯ 23)¯ (5)¯ (5) (32¯ 10)¯ (01¯ 23¯ 4)¯ (432¯ 10)¯ −ψkl ψij ψkl ψij + ψkl ψij ψkl ψij − ψkl ψij .

(a¯b... ) (a) (b)∗ We use the notation ψij = ψij ψij ··· to make the where the first term in the sum, n = 0, is simply |∅i. commutator slightly friendlier. Therefore, one can verify that Z = hground|groundi is The operators O±(Π, Γ) are built from the components precisely the MacMahon function (3). Γi,j and Πi,j. We can define O+(Π, Γ) by locally adding The Hamiltonian (20) can be thought of as the limit and removing a box in all possible vector positions (j, k), J0 = 0 of the more general model whilst O−(Π, Γ) is defined by locally removing and adding X ∗ 0 a box in all possible vector positions (j, k). Hhexagon = J0 ψ (~r)ψ(~r ) + H, (22) In AppendixB we build states corresponding to 2- and h~r,~r0i 3-boxes plane partitions and we also describe the action of the operators Γ, Π and O±(Π, Γ). Using this construction, where the first term describes nearest-neighbor interactions. one can rewrite the crystal melting Hamiltonian (2b) as In other words, the Hamiltonian (20) is an interaction Hamiltonian of 6th and 12th degrees, and if we assume a X H = − J Γj,k + Πj,k+ slightly more general setting with lowest-order hopping, j,k we obtain the Hamiltonian (22). (20) X √ 1 + V q Π Γ + √ Γ Π . j,k j,k q j,k j,k j,k V. TENSOR PRODUCT REPRESENTATION

The ground state of this model has been defined in [9]. It In the remainder of this work, we try to understand is the sum of the states corresponding to plane partitions |Λ| some universal features of the Hamiltonian (2b). In other Λ weighted by q , where |Λ| = # Boxes(Λ). In the words, we investigate aspects that do not depend on the notation above we have lattice descriptions above, in particular, its Hilbert space X |Λ| and underlying symmetries. The fact that the states in |groundi = q 2 |Λi our system are labeled by plane partitions suggests an Λ interpretation in terms of the MacMahon representation n X X Y n (21) 2 of the affine Yangian of glb(1) [20–24]. In this part of the = q Γja,ka |∅i , n paper we start our program of rewriting and interpreting n (j1,j2,...,jn)∈Z a=0 n (k1,k2,...,jn)∈Z the Hamiltonian (2b) in terms of Yangian invariants, which 6 we believe to be the key to understand the integrability in the tensor product a-slot. This condition clearly violates properties of the problem. We begin by identifying a the constrained growth of plane partitions; therefore we nontrivial coproduct structure of the operators that we impose a nontrivial coproduct ∆ : A → A ⊗ A by have defined in the previous sections. R (γ) := 1 ⊗ · · · ⊗ M(a−1)(γ) ⊗ R(a)(γ) ⊗ · · · ⊗ 1 , (28b) Plane partitions can be written as stacks of interlaced a integer partitions [9, 25], that is where the operator M(a−1)(γ) imposes that the a-slot growth does not violate the plane partition conditions, Λ = {Λ(1), Λ(2),..., Λ(N)|Λ(a) Λ(a+1)} , where and its eigenvalues are +1, if the a-slot integer partition Λ(a) = (λ(a), λ(a), . . . , λ(a)) , λ(a) ≥ λ(a) (23) grows consistently, and 0 otherwise. The existence of this 1 2 à i i+1 nontrivial coproduct is the first sign of a quantum group ∀ i = 1, 2, . . . , à , a = 1, 2,...,N − 1 . structure underlying the tensor representation above. As an instructive example, consider the configuration The number of boxes in the plane partition Λ is given by E | 2 i, with Γ| 2 i = | 2 i+ +| 3 i. In the tensor product 2 N notation we have X (a) |Λ| = #Boxes = |Λ | , (24) 3 a=1 M Γ| 2 i = Ra(γ)(| i ⊗ | i ⊗ |∅i) . (29a) in other words, it is the sum of squares of each integer a=1 partition layer. In this notation, a generic state |Λi asso- The action of R1(γ) reads ciated to a given plane partition Λ can be represented as (1) the tensor product R1(γ)(| i ⊗ | i ⊗ |∅i) = R (γ)| i ⊗ | i ⊗ |∅i E (1) (2) (N) |Λi = |Λ i ⊗ |Λ i ⊗ · · · ⊗ |Λ i , (25) = | i + ⊗ | i ⊗ |∅i E and the empty plane partition is the product of empty ≡ | 2 i + 2 integer partitions (29b) |∅i = |∅i ⊗ |∅i ⊗ · · · . (26) since the first layer is unconstrained. On the other hand

As we discuss in AppendixA, the usefulness of this R2(γ)(| i ⊗ | i ⊗ |∅i) = 0 , (29c) structure is that the integer-partition layers can de since any growth of the second layer violates the plane described in terms of free fermions with Neveu-Schwarz partition conditions. Finally boundary conditions. R3(γ)(| i ⊗ | i ⊗ |∅i) = As we have seen before, the operator Γ adds a box in all = | i⊗ M(2)(γ)| i ⊗ R(3)(γ)|∅i consistent places of a given partition Λ. Let us denote the + space of allowed places as Q (Λ), therefore = | i⊗| i ⊗ | i ≡ | 3 i . X (29d) Γ|Λi = |Λ + i . (27) More generally, the first layer can always grow freely as ∈Q+(Λ) long as it satisfies the integer partition rules. Using the Additionally, the operator Γ can be described in terms of equivalence between 2D Young diagrams and free fermions, (1) its action on the integer partitions: one can represent R (γ) as (1) X ∗ N ! R (γ) ≡ γ = Ψr+1Ψr . (30a) O (b) 1 Γ|Λi ≡ ∆(γ) |Λ i ⊗ |∅i r∈Z+ 2 b=1 (28a) Consequently, N+1 N ! M O (b) = Ra(γ) |Λ i ⊗ |∅i , R1(γ) = γ ⊗ 1 ⊗ · · · ⊗ 1 . (30b) a=1 b=1 In fact, γ acts as where γ is the growth operator on integer partitions. In γ|Λ(a)i = |(λ(a) + 1, λ(a), . . . , λ(a))i + ··· the definition above, for each layer labeled by a there 1 2 à exists an action of the integer partition growth operator (a) (a) (a) + |(λ1 , λ2 , . . . , λ + 1)i+ (30c) γ ∈ A, where A is the algebra of operators acting on the à + |(λ(a), λ(a), . . . , λ(a), 1)i . space of integer partitions H(a). In other words, given that 1 2 à |Λi ∈ N H , the operator Γ can be represented as the a (a) For the subsequent layers, the partitions are subject to the coproduct ∆(γ). Moreover, we assume a decomposition interlacing constraint, then ∆(γ) = L R (γ) that denotes the growth of each integer a a partition level. M(a−1) ⊗ R(a) |Λ(a−1)i ⊗ |Λ(a)i A trivial action of the coproduct ∆(Γ) is given by (a) (a−1) X (a) (31) Ra(γ) = 1 ⊗ · · · ⊗ R (γ) ⊗ · · · ⊗ 1, where the opera- = |Λ i ⊗ |Λ + i , a ≥ 2 , (a) tor R (γ) is the (representation of the) growth operator (a) (a−1) Λ +≺Λ 7

˜ (a) (a) which means that the resulting partition (Λ ) ≡ (Λ +) For the precedent layers, the partitions are subject to the ˜(a) (a−1) interlacing constraint, therefore satisfies λj ≤ λj ∀ j = 1, . . . , à. The first, and almost trivial, observation is that the operators M act diagonally, whilst R(a) is a constrained growth operator. S(a)⊗N (a+1) |Λ(a)i ⊗ |Λ(a+1)i We propose the form X (a) (a+1) = |Λ − i ⊗ |Λ i , a < N . (a−1) (a) X (a−1) (a) Λ(a)− Λ(a+1) M ⊗ R = Mi ⊗ Ri , (32) i (39)

(a) We propose the form where Ri acts as X (a) (a+1) (a) (a) (a) (a) (a+1) S (π) ⊗ N = Si (π) ⊗ Ni , (40) Ri |(λ1 , . . . ,λi ,... )i = (33) i (a) (a) |(λ1 , . . . , λi + 1,... )i. where N (a+1) is defined by The operator M(a−1) tests whether the partition (Λ(a) + ) i (a+1) (a+1) (a+1) (a−1) interlaces (Λ ), and it can be defined as Ni |(λ1 , . . . , λi ,... )i = (41a) (a+1) (a+1) si,a|(λ1 , . . . , λi ,... )i , M(a−1)|(λ(a−1), . . . , λ(a−1),... )i = i 1 i (34a) (a−1) (a−1) where i = 1, . . . , à, à and ri,a|(λ1 , . . . , λi ,... )i , ( 0 if λa − 1 < λa+1 where i = 1, . . . , à, à + 1 and s = i i . (41b) i,a 1 otherwise ( 0 if λa + 1 > λa−1 r = i i . (34b) i,a 1 otherwise It would be interesting to study the fermionic representation of the operators M and N and the quantum group Observe that the operator M(a−1) is nonlocal, since it acts structure of the operators Γ and Π in this context. On on the (a − 1)-slot but additional information on the states the other hand, in the next section we will see that the of the a-slot is required. It would very be interesting to growth of plane partitions is better described in terms of find representation for the operators M(a−1) ⊗ R(a), for the Yangian generators, and in this new context some of a ≥ 2, using the free fermions formalism. the underlying algebraic structure becomes more transpar- ent. It is worth mentioning that although the Yangian Similarly, the operator Π removes a box in all consistent algebra has its defining coproduct structure, it is different places, denoted by Q−(Λ), in a given partition Λ. Then from the coproduct structure we consider in this section. X Π|Λi = |Λ − i . (35) − VI. FERMION-BOSON DUALITY & AFFINE ∈Q (Λ) YANGIAN As before, we write The important observation of the previous section is N ! O the nontrivial coproduct structure of the operators Γ and Π|Λi ≡ ∆(π) |Λ(b)i ⊗ |∅i Π in the tensor product representation. In this section, b=1 (36a) we study other algebraic structure of these operators in N N ! M O (b) terms of the affine Yangians of gl(1), and we also define = Sa(π) |Λ i ⊗ |∅i , a fermion-boson correspondence for the plane partition a=1 b=1 states. where π acts on integer partitions. In addition In AppendixA we review the necessary tools we use in this section. In order to define a fermion boson- (a) (a+1) correspondence for plane partition states, let us write the Sa(π) = 1 ⊗ · · · ⊗ S (π) ⊗ N (π) ⊗ 1 · · · ⊗ 1 . (36b) states |Λi, that we call fermionic plane partition state, as The final layer can shrink freely as long as it satisfies N ~ (a) the integer partition rules, therefore O X χ (a) [C(k )] |Λi = Λ |~k(a)i z~k(a) (N) X ∗ a=1 ~k(a) S (π) ≡ π = ΨrΨr+1 (37) (42a) 1 ~ r∈Z+ X χΛ[C(K)] 2 = |K~ i, ZK~ and K~

π|Λ(a)i = |(λ(a) − 1, λ(a), . . . , λ(a))i + ··· where we use 1 2 à (38) + |(λ(a), λ(a), . . . , λ(a) − 1)i. |K~ i = |~k(1)i ⊗ · · · ⊗ |~k(N)i, (42b) 1 2 à 8 and where L~ is the conjugacy class associated to the plane partition Λ(N ) = Λ(N ) + . ~ ~ (1) ~ (N) L~ K~ χΛ[C(K)] = χΛ(1) [C(k )] ··· χΛ(N) [C(k )] (42c) Similarly, is a product of characters of the symmetric group. Using χ [C(J~)] X ~ X Λ− the results of AppendixA and equation (24), the number Π|Λ(NK~ )ii = χΛ[C(K)] |Λ(NJ~)ii Z ~ of boxes in Λ is simply Λ − ~ J ∈Q (Λ),J X N ≡ A− Λ(N ~ ) 7→ Λ(N ~ ) − |Λ(N ~ ) − ii , X X (a) K K K |Λ| = jk . (43) − j ∈Q (Λ(NK~ )) a=1 j (48a) ~ P P The states |ki can be labeled in terms of partitions where |Λ(N ~)| = |Λ(N ~ )| − 1 and ~ = − . J K J ∈Q (Λ(NK~ )) themselves, that is The amplitudes are ~ |ki = | ..., 2,..., 2, 1,..., 1i , (44) A Λ(N ) 7→ Λ(N ) − = | {z } | {z } − K~ K~ k2 times k1 times ~ X X χΛ− [C(J)] (48b) = χ [C(K~ )] , Λ Z and we denote these states as |~ki ≡ |N ii. For example, − J~ ~k Λ ∈Q (Λ) the state associated to the vector ~k = (0, 1, 2, 1, 0, 0,... ) is 2 ~ α−4α−3α−2|0i and we denote it as a bosonic plane partition where J is the conjugacy class associated to the plane state |(4, 3, 3, 2)ii. Using the Frobenius formula (A10a), we partition Λ(NJ~) = Λ(NK~ ) − . We give the amplitudes can write the plane partition state in terms of |N~k(a) ii, for the 0-, 1- and 2-box transitions in AppendixD. Working with the (bosonic) plane partition states |NK~ ii N ~ (a) seems to be much more involved, but it, in fact, has some O X χΛ(a) [C(k )] |Λi = |N~k(a) ii conceptual advantages. In particular, the action of the z~k(a) a=1 ~k(a) growth operators on |N ii suggests a relation between (45) K~ ~ these states and the MacMahon representation of the affine X χΛ[C(K)] = |Λ(N )ii . K~ Yangian of glb(1) [20–24]. More specifically, we have the ZK~ K~ following conjecture: Its inverse is X The growth operators Γ, Π and O±(Γ, Π) are linear |Λ(N )ii = χ [C(K~ )]|Λi . K~ Λ (46) combinations of the generators of the algebra Y[glb(1)], Λ and the states |NK~ ii fulfill the Y[glb(1)] MacMahon The relations (45) and (46) define the plane partition representation. fermion-boson correspondence. From (43), it is clear that

|Λ| = |Λ(NK~ )|, therefore, the correspondence between the plane partition basis preserves the number of boxes The algebra Y[glb(1)] is an associative algebra defined grading. It can be seen form the fact that the only nontrivial by the generators {ej, fj, ϕj | j ≥ 0} and a set of (anti-) characters associated to the Young diagram Λ are those commutations that can be found in [20]. One of the most related to the conjugacy classes of |Λ|. In AppendixC we important features of the affine Yangian of glb(1) is its write these relations for a few states. triangular decomposition Y[glb(1)] = Y+ ⊕ B ⊕ Y− which We would like to represent the growth operators, and is generated, respectively, by ej, ϕj and fj. consequently the Hamiltonian (2b), in terms of the new As a first evidence in favor of our conjecture, it has been bosonic basis |Λ(N~k(a) )ii. It is easy to see that since the shown that in the MacMahon representation studied by map preserves the number of boxes Procházka [20], the operators ej and fj act precisely as in (47a)-(48a), that is χ [C(L~ )] X ~ X Λ+ Γ|Λ(NK~ )ii = χΛ[C(K)] |Λ(NL~ )ii X ZL~ ej|Λii = Ej(Λ 7→ Λ + )|Λ + ii , Λ ∈Q+(Λ),L~ ∈Q+ X (49) ≡ A+ Λ(N ~ ) 7→ Λ(N ~ ) + |Λ(N ~ ) + ii , X K K K fj|Λii = Fj(Λ 7→ Λ − )|Λ − ii. ∈Q+(Λ(N )) K~ ∈Q− (47a) Furthermore, the character of this representation is pre- P P where |Λ(N~ )| = |Λ(N ~ )| + 1 and ~ = + . cisely the MacMahon function (3), and the operators ϕ L K L ∈Q (Λ(NK~ )) j Therefore, we have the amplitudes act diagonally as A+ Λ(NK~ ) 7→ Λ(NK~ ) + = ϕj|Λii = ϕj,Λ|Λii . (50) ~ X X χΛ+ [C(L)] (47b) = χ [C(K~ )] , Finally, if the conjectured correspondence between the Λ Z + L~ Λ ∈Q (Λ) generators of Y[glb(1)] and with the growth operators Γ 9 and Π is indeed correct, we immediately conclude that We should also observe that the maps relating the O±(Π, Γ) ∈ B since they can be written as linear combina- Heisenberg and the Yangian generators are of the form tions of ϕ . In other words, both systems are precisely the α ∝ adm−1e and α ∝ adm−1f ; and that the Heisen- j −m e1 0 m f1 0 same once we identify the amplitudes A± with linear com- berg subalgebra is the only ingredient we used in the defini- binations of the coefficients Ej and Fj. We then conclude tions of the bosonized expressions. Therefore, it is possible that the Hamiltonian (2b) can be written as that the only nontrivial coefficients in (51) are a0 and a1. Finally, we have specifically realized the construction in X terms of the linear W , but one can try to generalize H = −J aj (ej − fj) 1+∞ it to the nonlinear W domain assuming, for example, j∈Z+ 1+∞ (51) that the Hamiltonian (51) is the defining problem. X √ 1 + V q b + √ c ϕ + V (c ϕ + c ϕ ) , In conclusion, despite the evidence in favor of the inte- j q j j 0 0 1 1 j∈Z+ grability of the Hamiltonian (2b), we have not yet been able to close this question. Let us hope nature does not where the coefficients aj, bj and cj are necessary to define disappoint us. the equivalence, including combinatorial terms, without introducing new constraints to the Yangian Y[glb(1)]. Ob- serve that we write the generators ϕ0 and ϕ1 separately VIII. ACKNOWLEDGMENTS since they are central elements. T.A. is supported by the Swiss National Science Founda- tion under Grant No. PP00P2 183718/1. D.O. acknowl- VII. CONCLUSIONS AND OUTLOOK edges partial support by the NCCR 51nf40–141869 “The Mathematics of Physics” (Swissmap). The work of S.R. is In this work we have studied the 3D quantum crystal supported by the Swiss National Science Foundation under melting of [9], governed by the Hamiltonian (2b) in various Grant No. PP00P2 183718/1. guises. An unanswered question is whether or not this system is quantum integrable, as is the case for its 2D cousin. In the current paper, we push this study further. Appendix A: 2D Fermion-Boson Duality We have rewritten the dimer model as a fermionic system with 6th-order interactions in a Kagome lattice. We can Bosonization in two dimensions can be rephrased as a think of this relation as a duality, where the extended duality between the fermionic and bosonic Hilbert spaces. object (the dimer) in the hexagonal lattice corresponds to In this section we would like to review the necessary in- a particle in the dual Kagome lattice. This description and gredients for the construction of the fermion-boson duality its possible advantages have not yet been fully explored. for plane partitions of SectionVI, in this section we fol- Using the tensor product representation of the plane low [3,7–9, 26, 27]. partitions, we have argued that the growth operators have Neveu-Schwarz free fermions states can be classified in a nontrivial coproduct, which may point to the existence terms of integer partitions. Using the algebra of underlying quantum group structures. In fact, it is ∗ ∗ interesting to explore the conjectured relation between this {ψm, ψn} = {ψ , ψ } = 0, m n (A1) structure and the coproduct of the affine Yangian of glb(1). {ψ∗ , ψ } = δ , Additionally, using the fermion-boson correspondence, we m n mn have been able to extend the duality to the 3D plane parti- the fermionic state |λi in the Hilbert space H is given by tion states. Putting all these facts together, we managed to write the crystal melting Hamiltonian in terms of Yangian r Y generators, although the conjecture is yet to be proved. |λi = ε ψ ψ∗ |0i , (A2) −mi−1/2 −ni−1/2 i=1 There is a myriad of interesting problems still to be addressed. As mentioned, the Kagome lattice description where the vacuum |0i is defined by the conditions is still poorly understood and it may be an interesting description from the viewpoint of condensed matter physics. ∗ ψn|0i = ψn|0i = 0 ∀ n > 0 , (A3) It would be very desirable to study the dynamics of the Pr more general Hamiltonian (22), and see how the states and ε = (−1) ni+r(r−1)/2. classified by plane partitions emerge from the limit J0 = 0. One can now associate the plane partition λ = We have not specified details of the Yangian Y[glb(1)], (λ1, λ2,... ) to the state above, which in Frobenius co- but we should mention that we have been able to use its ordinates reads representation because there are isomorphisms relating the Yangian Y[glb(1)], the W1+∞ and the spherical Hecke alge- λ = {(mi|ni)|i = 1, . . . , r} . (A4) c bra SH , see [20, 21]. More specifically, the linear W1+∞ has a well-known representation in terms of free fermions, The number of boxes associated to this Young diagram is and in Section (VI) we have used the Heisenberg subal- X X gebra uˆ(1) ⊂ W1+∞ to define the bosonic plane partition |λ| = λj = (mi + ni) . (A5) states. j i 10

In two dimensions, one can expand the same Hilbert (0, 1, 0,... ) (2, 0,... ) space H in terms of a chiral bosonic ﬁeld ∂φ(z) deﬁned by the components χ(1,1) −1 1 χ(2) 1 1 X • ∗ • αm = • ψ−jψj+m • , (A6a) j∈Z+1/2 (0, 0, 1,... ) (1, 1, 0,... ) (3, 0, 0,... ) which satisfy the Heisenberg algebra χ(1,1,1) 1 −1 1

[αm, αn] = mδm+n,0 . (A6b) χ(2,1) −1 0 2 χ 1 1 1 Using these operators, states in H are of the form (3)

∞ ~ Y kj |ki ≡ |k1, k2,... i = (α−j) |0i . (A7) Appendix B: Kagome lattice Hamiltonian j=1 In this section we use the operators Γ and Π to argue that ~ The relation between the two bases |ki and |λi is given the Hamiltonian (20) in the Kagome lattice is equivalent by to (2b). ~ Let us first write some states. The 2-boxes configurations X χλ[C(k)] |λi = |~ki , (A8a) are z~ ~ k E k = Γ1,0| i, | i = Γ0,1| i, | 2 i = Γ−1,−1| i , and its inverse is (B1a) ~ X ~ and inversely, |ki = χλ[C(k)]|λi , (A8b) E λ | i = Π1,0 , | i = Π0,1 | i , | i = Π−1,−1| 2 i . Q∞ kj ~ (B1b) where z~ = kj!j , and χλ[C(k)] is the character of k j=1 Additionally, the 3-box configurations are the symmetric group Sn evaluated in the representation λ and conjugacy class C(~k). Moreover, the number of boxes E in the partition λ is the level of the state |~ki, that is = Γ2,0 , | i = Γ0,2 | i ,

X |λ| = jk . (A9) j E E j≥1 = Γ0,1 = Γ1,0 | i , The characters can be easily calculated using the Frobe- nius character formula [28] | 3 i = Γ−2,−2| 2 i , (B2a)

 r  Y k E E χ [C(~k)] = ∆(~x) P (~x) j , (A10a) = Γ , | 2 i = Γ | i λ  j  2 −1,−1 −1,−1 j=1 (` ,...,` ) 1 m and inversely, with E E E m = Π2,0 = Π0,1 = Π−1,−1 2 , Y X j ∆(~x) = (xi − xj) ,Pj(~x) = xi . (A10b) i

`1 `m in the dual lattice. f(~x)|(` ,...,` ) := coeﬀ. of x1 ··· xm . (A12) 1 m In summary, we write the kinetic term of the Hamiltonian For completeness, we give here the character tables we (2b) as use in the text: X X X | ih | + | ih | = Γj,k + Πj,k . (B4) χ(1)[C(1, 0,... )] = 1 , (A13) j,k 11

and the 3-box states | i and | 3 i, work similarly. Additionally, E E X Πi,jΓi,j = 4 i,j E E (B11) X Γi,jΠi,j = 2 , FIG. 13: A 5-box configuration i,j E and the states | 2 i and work similarly. 2 For the potential term, we need to write operators that Using this logic, one may see that the Hamiltonian (20) give the number of places where we can add a box, and the is equivalent to (2b). number of boxes that can be consistently removed. These operators can be written as products of Γ and Π. For example, given the empty configuration, it is easy to see Appendix C: Basis transformation that X X Γ Π |∅i = 0 , Π Γ |∅i = |∅i , (B5) Using the characters (A13) we can find the basis trans- i,j k,l i,j k,l formation (45- 46) for some states. For example, the 1-box i,j;k,l i,j;k,l state is simply that, naturally, means that we have one available place to ⊗ add a box, and we do not have any box to be removed. | i = |(1)ii⊗ |∅i ≡ | ii . (C1a) For two boxes, we see that there must be relations between the two sets of indices (i, j) and (k, l). Initially we The states with two boxes are have 1 E | i = | ii + i (C1b) X 2 Γ0,0Πi,j| i = | i , (B6) i,j E 1 E = − | ii− i and it is obvious that we have 1 box to be removed. Fur- 2 thermore,

X 2 ⊗ 2 | i =Π1,0 Γj,k| i = Π1,0Γ1,0| i | i = |(1)ii⊗ |(1)ii⊗ |∅i = | ii j,k∈Z X and the states with three boxes are =Π Γ | i = Π Γ | i 0,1 j,k 0,1 0,1 (B7) j,k∈Z 1 1 E 1 | i = | ii + i + i (C1c) X 3 2 6 =Π−1,−1 Γj,k| i = Π−1,−1Γ−1,−1| i . j,k∈Z 1 1 E 1 Therefore = | ii− i + i 3 2 6 X Πi,jΓi,j| i = 3| i , (B8) i,j E 1 = − | ii− i and now we have 3 places to add a box to this conﬁguration. 3 It is straightforward to see that this logic remains the same for other conﬁgurations. Furthermore, we also have 1 E | 2 i = | 2 ii + i 2 E E 2 X Πi,jΓi,j = 3 , i,j E 1 E E E (B9) = − | 2 ii− i X 2 2 Γi,jΠi,j = , 2 i,j

| 3 i = | 3 ii . and the other 2-box states, | i and | 2 i, work similarly. For 3 boxes we have two distinct cases. The ﬁrst case is The nontrivial inverse relations are X E Πi,jΓi,j = 3 | ii = | i − (C2a)

i,j (B10) X Γi,jΠi,j = E E i = | i + i,j 12 and It is also immediate to see that the transition 1 7→ 0-boxes E is

| ii = | i − + (C2b) A−[ 7→ ∅] = 1. (D2e) E Therefore i = | i + 2 + E Γ| ii = i + || 2 ii (D2f) E Π| ii = |∅ii . i = | i −

We can repeat the analysis above to consider the tran- and ﬁnally sitions 2 7→ 3 boxes and 2 7→ 1 box. These are given E by | 2 ii = | 2 i − (C2c) 2 A+[ 7→ ] = 0 A−[ 7→ ] = 0 E E h i i = | 2 i + . A+ 7→ = 1 (D2g) 2 2 A+[ 7→ 2 ] = 1

Appendix D: Amplitudes h i Using formulas (47b) and (48b) we can ﬁnd the ampli- A+ 7→ = 1 A− 7→ = 2 tude A for the ﬁrst states. It is straightforward to see ± h i (D2h) that A+ 7→ = 0 h i A+[∅ 7→ ] = 1, A−[∅ 7→ 0] = 0 , (D1) A+ 7→ 2 = 1 where χ∅(C[~0]) = 1. The transitions 1 7→ 2 boxes are

1 A+ [ 2 7→ 2 ] = 0 A− [ 2 7→ ] = 1 A+[ 7→ ] = χ(1)[C(1, 0,... )] Z(0,1,0... ) h i A 2 7→ = 1 (D2i) + 2 × χ(2)[C(0, 1,... )] + χ(1,1)[C(0, 1, 0 ... )] A+ [ 2 7→ 3 ] = 1. (D2a) + χ(1);(1)[C((0, 1, 0 ... ); (0, 0,... ))] 1 = (1 − 1 + 0) = 0 , 2 where we have used (42c) to see that

χ(1);(1)[C[(0, 1, 0,... )] (D2b) = χ(1)[C[(0, 1, 0 ... )]χ(1)[C[(0,... )] = 0 . Similarly, h i 1 A+ 7→ = χ(1)[C(1, 0,... )] Z(2,0,... ) × χ(2)[C(2, 0,... )] + χ(1,1)[C(2, 0,... )] (D2c) + χ(1);(1)[C((2, 0,... ); (0, 0,... ))] = 1 , and 1 A+[ 7→ 2 ] = χ(1)[C(1, 0,... )] Z(1,0,... );(1,0,... ) × χ(2);(∅)[C(1, 0,... ); (1, 0,... )]

+ χ(1,1);(∅)[C(1, 0,... ); (1, 0,... )] + χ(1);(1)[C((2, 0,... ); (0, 0,... ))] = 1 . (D2d) 13

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